With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
In a category with all products, one can take finite products of objects, and hence one has a monoidal category. One can however also take infinite products. In some situations, such as in categorical probability, one is interested in taking an infinitary version of a tensor product.
In measure-theoretic probability theory, such a construction is provided by Kolmogorov's extension theorem, and the same idea can be used in general:
Let $(C,\otimes,I)$ be a symmetric monoidal category equipped with a distinguished map $del_X:X\to I$ for each object $I$. (This happens for example if $C$ is cartesian semicartesian, or if it has a Markov or copy-discard structure.)
Let $J$ be a possibly infinite set, and let $(X_i)_{i\in J}$ be a $J$-indexed collection of objects of $C$.
For every finite subset $F\subseteq J$, denote by
the tensor product of the family $(X_i)_{i\in F}$. (Note that $F$ is not ordered, but the ordering of the terms in the tensor product does not matter up to coherent isomorphism, since $C$ is symmetric monoidal.)
Consider now finite subsets $F\subseteq G$ of $J$. We have a canonical “projection” morphism defined as follows. First of all, for $i\in G$, let
Similarly, let $e_i:X_i\to Y_i$ be
The map $\pi_{G,F}$ is given by the tensor product of the $e_i$ as follows, where the isomorphism on the right is given by the unitors of the monoidal category.
Let now $Fin(J)$ be the poset of finite subsets of $G$. The union of finite subsets is finite, so $Fin(J)$ is a directed set, hence a filtered category. The functor $Fin(J)^op\to C$ mapping the inclusion $F\subseteq G$ to $\pi_{G,F}$ defined above is hence a cofiltered diagram.
We say that an object is an infintary tensor product of the family $(X_i)_{i\in J}$ in $C$, and denote it by $X_J$, if
In a cartesian monoidal category, every infinite product, if it exists, is an infinitary tensor product.
More generally, consider a cartesian monoidal category $\mathcal{C}$ and a monoidal monad (or commutative monad) $T$. Recall that the Kleisli category of a monoidal monad is canonically a monoidal category. Suppose now that an infinite product in $\mathcal{C}$ exists, and express it as a cofiltered limit of finite products. The left adjoint $\mathcal{C}\to\mathcal{C}_T$ preserves this limit if and only if the endofunctor $T$ does (see here). So, given a category with infinite products and a monoidal monad on it which preserves cofiltered limits, its Kleisli category has infinitary tensor products.
In a Markov category, Kolmogorov products are particular infinitary products compatible with the copy-discard structure. In particular, the category BorelStoch has all countable infinitary tensor products.
The Giry monad on standard Borel spaces is an instance of both of the examples above, by the Kolmogorov extension theorem.
Reversing all arrows, the infinite tensor product of rings is defined as the filtered colimit of all the finitary tensor products. (The canonical arrows $I\to X$ are the units of the rings.)
George Janelidze and Ross Street, Real sets, Tbilisi Mathematical Journal, Vol. 10, No. 3, 2017, pp. 23-49. [doi:10.1515/tmj-2017-0101]
Tobias Fritz and Eigil Fjeldgren Rischel, Infinite products and zero-one laws in categorical probability, Compositionality 2(3) 2020. (arXiv:1912.02769)
Last revised on September 28, 2024 at 17:40:26. See the history of this page for a list of all contributions to it.